I shoot 6x6, 6x9 and 4x5. I would love to say that I choose the format that best suits the subject, but that would be a lie. I find that the square is more demanding. It is far less dramatic than the 6x9 (same as 35mm) and less formal, to my eye, than 4x5. The square is less forgiving of unused space. By its very nature, every area is equal and requires attention.
When I compose w/ the square I don't think in thirds spatially but maybe in importance. In that I will put something of lesser importance (~33%) on one side and some thing of greater importance (~66%) on the other. Or use that type of balance front to back. Another technique is to have the foreground lead you to the subject. The final technique I seem to employ is to stuff the frame with the object. This stuffing causes the square to become a container that just barely holds the image. In all formats negative space is important and is possibly most important in a square.